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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 1

Impact of Interbranch Correlation on Multichannel

Spectrum Sensing with SC and SSC Diversity

Combining Schemes

Salam Al-Juboori, Student Member, IEEE, Xavier Fernando, SMIEEE, Yansha Deng and

A. Nallanathan Fellow, IEEE,

Department of Electrical and Computer Engineering, Ryerson University, Toronto, Canada

Department of Informatics, King’s College London, London, WC2R 2LS, UK

School of Electronic Engineering and Computer Science, Queen Mary University of London, U.K

email: saljuboo@ryerson.ca, fernando@ee.ryerson.ca, yansha.deng@kcl.ac.uk, a.nallanathan@qmul.ac.uk

Abstract—Multi antenna receivers are often deployed in

cognitive radio systems for accurate spectrum sensing. However,

correlation among signals received by multiple antennas in these

receivers is often ignored which yield unrealistic results. In

this paper, the effect of this correlation is accurately quantiﬁed

by deriving analytical expressions for the average probability

of detection. Alternative simpler expressions are also derived.

These are done for Selection Combining (SC) and Switch and

Stay (SSC) diversity techniques in dual arbitrarily correlated

Nakagami-mfading channels. Then it is repeated for triple

exponentially and identically correlated Nakagami-mfading

channels with SC diversity technique. Analysis results show that

the inter-branch correlation impacts the detector performance

signiﬁcantly, especially in deep fading scenarios. Also, SC

outperforms SSC as expected. However, the difference between

them becomes very small in low fading and highly correlated

scenarios which, indicates that the simpler SSC scheme can as

well be deployed in such situations.

Index Terms—Cognitive radio networks, spectrum sensing,

inter-branch correlation, diversity combining, selection

combining, switch and stay combining.

I. INTRODUCTION

P

ROTECTING the primary users from detrimental

interference from the secondary user signals is crucial in

cognitive radio systems. Accurate spectrum sensing is essential

for this. Simple schemes such as Energy Detectors (ED) are

widely used for this purpose that detect weak signals in noisy

channels as long as the noise power is known [1]. Accurate

spectrum sensing suffers from few issues, multipath fading and

shadowing being the leading causes. Multi antenna receivers,

with appropriate diversity combining schemes, are designed

to overcome these issues. Ideally, wireless channels seen by

the multiple antennas shall be independent to obtain the best

results from these diversity receivers [2]. However, often this is

not the case especially, when antennas are increasingly placed

closer to each other as the mobile units get smaller and more

demanding. Therefore, ignoring inter-branch correlation yields

inaccurate, especially overly optimistic, results. The effects

of multipath fading and correlation among antenna branches

heavily depend on the type of the diversity combining technique

employed. It is well known that Maximal Ratio Combining

scheme (MRC) is the optimal scheme which is also the

most complex linear diversity scheme. Equal Gain Combining

(EGC) diversity technique is a close competitor. Both the

MRC and EGC techniques require all or some knowledge

of the Channel State Information (CSI) [3]–[5]. Furthermore,

in these schemes each diversity branch must be equipped

with a single receiver that increases the system complexity.

Recently simpler combining schemes such as Switch and Stay

Combining (SSC) and Selection Combining (SC) are getting

popular due to their simplicity. These are especially useful in

cognitive radio networks. With the SC scheme, the receiver

simply selects the antenna with the highest received signal

power and ignores other antennas. Hence, signal combiners,

phase shifters or variable gain controllers are not required [3],

[6]. The SSC diversity technique is the least complex system

where no real combiner is required. The SSC selects a particular

antenna branch until its SNR drops below a predetermined

threshold [3]. Both SC and SSC schemes are required to

measure only the amplitude on each branch (in order to select

the highest one). Hence, they can be employed for both coherent

and non-coherent modulation schemes [3]. Different diversity

combining techniques have been studied in the literature. In [7],

averaging the probability of detection over fading channels with

Rayleigh, Nakagami-

m

and Rician distributions are studied, and

closed-form expressions for detector parameters were derived

for Nakagami-

m

channels with integer values of

m

. In [8]-[9],

alternative analytic approaches to that in [1] and [7] were given.

Furthermore, in [8], independent and identically distributed

(

i.i.d

) dual and

L

number of Rayleigh fading branches were

considered with SSC and SC diversities. Corresponding average

probability of detection expressions (

PD

) were also derived

for both techniques. In [10]-[11], closed-form expressions for

PD

were derived for

i.i.d.

diversity branches in Nakagami-

m

fading channels employing SC technique. In [12], closed-form

expressions of

PD

for

i.i.d

dual Nakagami-

m

fading branches

with SSC were derived for real and integer

m

values. In

our previous work [6], we have done an investigation on

the probability of detection for SC diversity with correlated

Nakagami-

m

fading branches. A review of prior works reveals

that correlated fading branches with SSC diversity is not studied

in the literature. However, because of the simplicity, the SSC

is particularly valuable for mobile stations that have limited

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 2

resource and power. This paper aims to fulﬁll that requirement.

In this paper, we extend our previous investigations by

considering SC and SSC diversity combining techniques with

identically correlated branches in Rayleigh and Nakagami-

m

fading channels.

Our contribution falls into two folds. First

•

We consider SC and SSC schemes with dual arbitrarily

correlated branches in Rayleigh and Nakagami-

m

fading

channels.

•

Then, we extend study of SC diversity to triple

exponentially correlated branches.

•

Corresponding novel expressions for average probability

of detection are derived for each case.

•

Alternative and more general and simpler expressions are

also derived for each case.

•

For SSC diversity, we derive an expression which can be

solved numerically to calculate the optimal SNR threshold

value in order to optimize the detector performance.

•All our derived expressions do converge rapidly.

Secondly, to gain better insight

•

We do a performance comparison between the two

combining diversity techniques

•

Analysis results show that the inter-branch correlation

affects the detector performance signiﬁcantly, especially

in deep fading scenarios.

•

SC outperforms SSC as expected however; the difference

between them becomes very small in low fading scenario

with highly correlation among antennas. This indicates

that the simpler SSC scheme can be substituted for the

SC scheme in these situations.

The rest of the paper is organized as follows. Section II

describes the system model. In section III, we study the

performance of SC scheme. In section IV, we study the

performance of SSC scheme. Section V describes simulation

and analysis results. Section VI concludes the paper.

II. SY ST EM MO DE L

We follow a binary hypothesis testing on the received signal

to declare the presence or absence of the primary user. For

this, we employ ED that is widely used in cognitive spectrum

sensing. Note that no priori information about the detected

signal is needed for ED [13], [14].

Let x(t)be the received observations data

x(t) = h s (t) + n(t),(1)

where,

h

is the complex channel gain amplitude coefﬁcient,

assumed to be constant during the sensing time,

s(t)

is the

signal to be detected and,

n(t)

is the AWGN noise. This noise

is a low-pass Gaussian process with zero mean and variance

N0W

where,

N0

and

W

denote Power Spectral Density (PSD)

of the Gaussian noise and the signal bandwidth, respectively.

Two hypotheses are deﬁned for the decision statistics.

Namely

H0

and

H1

, for the absence and the presence of

the primary user signal respectively, as follows:

x(t) = (n(t)under H0

h s (t) + n(t)under H1.(2)

The decision statistics is squared and integrated over time

T

at the ED. The output is written as

y,2

N0ZT

0|x|2(t) dt. (3)

The Probability Density Function (PDF) of the decision

statistics yis given by [8] and [9]

pY(y) =

1

2uΓ(u)yu−1e−y

2,under H0

1

2y

2γu−1

2e−2γ+y

2Iu−1√2γ y,under H1

(4)

where

γ

denotes the signal-to-noise-ratio,

Γ(.)

is the the

Gamma function and,

Iν(.)

is the

νth

order modiﬁed Bessel

function of the ﬁrst kind. The parameter

u

depends on the

time-bandwidth product. In (4), it is clear that the decision

statistics has a central chi-square distribution with

2u

degrees

of freedom

χ2

2u

in the absence of the primary user signal,

i.e. the received samples are noise only. However, it has a

non-central chi-square distribution

χ2

2u(ψ)

with

2u

degrees of

freedom and non-centrality parameter

ψ= 2γ

in the presence

of the primary user signal [8] and [9].

Let us deﬁne

λ

as the decision threshold. Then the probability

of false alarm (

PF

) and the probability of detection (

PD

) of

the ED can be written as

PF=P r (y > λ |H0),(5)

PD=P r (y > λ |H1).(6)

where

P r (.)

denotes the Cumulative Distribution Function

(CDF). Consequently, the probability of false alarm and

probability of detection in AWGN channel are given as [8] -

[9]

PF=Γu, λ

2

Γ(u),(7)

PD=Qup2γ, √λ,(8)

where

Γ(., .)

and

Qu(., .)

denote the upper incomplete Gamma

function and generalized Marcum Q-function, respectively.

These detection probabilities are conditioned upon the channel

realization. Also, they represent instantaneous probability of

detection. Therefore, we need to integrate this instantaneous

probability of detection over the SNR’s PDF of the

corresponding fading channel (

pγ Div (γ)

) to obtain the average

probability of detection PD,D iv1.

PD,D iv =Z∞

0

Qup2γ, √λpγ Div (γ) dγ . (9)

The expression in (9) will serve as a general expression for

the corresponding diversity channel.

Note that the probability of detection expression in (8) is

restricted to only integer values of

u

since the PDF of the

decision statistics in (4) is derived only for even numbers,

i.e.

2u

, as stated in [8]. However, when the alternative

Marcum-Q function is employed,

u

could be half-odd integer

1

False alarm probability is not a function of SNR as no signal is transmitted,

therefore it will remain unchanged as in (7).

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 3

(

u∈ {0.5,1,1.5,2,2.5,3, ...}

, i.e. not restricted to integer

values) [15]. Furthermore, the fading parameter

m

in Nakagami

channels might also be not restricted to integer values

depending on the mathematical method employed to solve the

integral in (9). This highlights the advantage of the alternative

expressions which we derive later using alternative Marcum-Q

function.

III. SC DIVERSITY WITH COR RE LATE D NAKAGAMI-m

FADING CHANNELS

In this section, we will derive the average probability of

detection for dual and triple Nakagami-

m

correlated fading

branches with SC diversity.

Selection Combining is a low complexity diversity technique,

as it chooses the highest SNR’s branch using the relationship

r=max {rl, l = 1,2, ... L}.(10)

Therefore it processes one branch a time. Consequently, no

phase knowledge is required.

The PDF of a univariate

r

-Nakagami-

m

variable is given

by [16]

fr(r) = 2

Γ(m)m

Ωmr2m−1e−m

Ωr2, r ≥0(11)

where,

Γ(.)

denotes the Gamma function,

Ω = E[r2]/m =¯

r2

m

is the mean value of the variable

r

, and

m

(

m≥1/2

) is the

inverse normalized variance of

r2

, which describes the fading

severity.

We deﬁne the instantaneous SNR per symbol per channel

γl

as

γl=r2

lEs

N0

;

l∈[1,2, ... L]

;

Es

is the energy per symbol

and,

N0

is the PSD of the Gaussian noise. The average SNR

per branch is

¯γl=¯

r2

l

Es

N0

where,

¯

r2

l=E[r2

l]

is the expectation

of the channel envelop.

A. SC with Dual arbitrarily Correlated Branches

Using [[17], Eq. (20)] and, by assuming identical diversity

branches and by changing variables with some mathematical

simpliﬁcation, the PDF of the output SNR for a dual SC

combiner under correlated Nakagami-

m

fading channels can

be obtained as

pγ SC (γ) = 2

Γ(m)m

¯γm

γm−1exp −m γ

¯γ

×h1−Qmp2a ρ γ, p2a γi, γ ≥0

(12)

where,

ρ

denote the correlation coefﬁcient between the two

fading envelopes, and

a=m

¯γ(1−ρ)

. Please see Appendix A for

detailed derivation.

By substituting (12) into (9), the average probability of

detection for dual correlated SC’s diversity branches (

PD,S C,2

)

is obtained as

PD,S C,2=2

Γ(m)m

¯γm

[IA−IB],(13)

where

IA=Z∞

0

Qup2γ , √λγm−1exp −mγ

¯γdγ, (14)

and

IB=Z∞

0

Qup2γ , √λQmp2aργ, p2aγ

×γm−1e−mγ

¯γdγ.

(15)

Note this lengthy expression consists of two integrals,

IA

and

IB. We solve them separately. Please see Appendix B.

Hence, the average probability of detection for dual

SC receiver under correlated identical Nakagami-

m

fading

branches (restricted to integer uand mvalues) is

PD,S C,2=2

Γ(m)m

¯γm"1

2m−1(G1+η

2

u−1

X

n=1

1

n!λ

2n

×1F1m;n+ 1; λ¯γ

2 (m+ ¯γ)−

∞

X

n=0

∞

X

i=0

i+m−1

X

k=0

Γu+n, λ

2

Γ(u+n)n!

×ai+kρi(i+k+m+n−1)!

ci+k+m+ni!k!#,

(16)

where

c=1 + a(ρ+ 1) + m

¯γ

and

G1

for integer

m

values

is

G1=2m−1(m−1)!

m

¯γ2m¯γ

m+ ¯γe−λ

2

m

m+¯γ

×"m+ ¯γ

¯γ m

m+ ¯γm−1

Lm−1−λ¯γ

2 (m+ ¯γ)

+

m−2

X

n=0 m

m+ ¯γn

Ln−λ¯γ

2 (m+ ¯γ)#.(17)

Here

Ln(.)

denotes Laguerre polynomial of

n

-degree [18],

and

1F1(., .;.)

denotes the Conﬂuent Hypergeometric function.

This is deﬁned in [[19], Eq. (15.1.1)] as

1F1(a1, b1;x) = Γ(b1)

Γ(a1)

∞

X

i=0

Γ(a1+i)xi

Γ(b1+i)i!.(18)

Note (16) reduces to dual correlated Rayleigh fading branches

for

m= 1

. It’s worthwhile to mention that for

i.i.d.

diversity

branches, (16) reduces to [[9], Eq.(7), [8], Eq. (20)] multiplied

by 2 (not exceeding unity)). The latter expression was derived

for the average probability of detection in ﬂat fading. Hence

we have improved the detection performance and derived (16)

to serve as a proof.

B. Alternative Expression for PD,SC,2

Despite the fact that

Qu√2γ, √λ

portion of the second

integral

IB

in (13) is evaluated for

u

values not-restricted to

integer, (16) is still restricted to integer values. This is because,

the ﬁrst integral

IA

in (13) is only valid for integer

u

and

m

values. In this section, we derive a more general and simpler

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 4

alternative expression for (16) that is not restricted to integer

uvalues. Please see Appendix C for the derivation.

PD,S C,2= 1 −2m

¯γm

e−λ

2

×"1

dm

∞

X

n=uλ

2n1

Γ(n+ 1)1F1m; 1 + n;λ

2d

−1

Γ(m)

∞

X

n=u

∞

X

i=0

i+m−1

X

k=0 λ

2nΓ(m+i+k)ai+kρi

Γ(n+ 1)cm+i+ki!k!

×1F1m+i+k; 1 + n;λ

2c#.

(19)

For

m= 1

, (19) reduces to the average probability of

detection with dual correlated Rayleigh fading branches and,

with

ρ= 0

to

i.i.d

dual Rayleigh fading branches given in (

[8], Eq. (30)).

Fortunately, the error resulting from truncating the

inﬁnite series in (19) is upper bounded by the Conﬂuent

Hypergeometric function deﬁned in (18). Since this function

is monotonically decreasing with

i, k

and

n

for given values

of

m

,

λ

and

¯γ

[20], the number of terms (

Nn

and

Ni

) that

required ﬁve digit accuracy could be calculated. These numbers

are shown in Table I for different values of ρand m.

It’s worthwhile to mention that several solutions for integrals

involving the Marcum Q-function are available in literature

[21]–[25]. However, our case of study in (15) solves a different

and more complicated integral which involves a product of

two Marcum Q-functions. These solutions are introduced in

(57), (69) in Appendices B and C, respectively. To the best of

knowledge, we believe that this solution is new in literature.

Finally, we’d like to mention that the solutions introduced

in expressions (16) and (19) present a clear advantage

over the numerical integration approach showed in (13)

since a numerical integration is rather long and often gives

approximated result. Furthermore, although expressions in

(16) and (19) involve nested inﬁnite series, they are either

upper bounded by a monotonically decreasing conﬂuent

hypergeometric function or by an upper incomplete gamma

function. Note that the latter could also be represented by a

monotonically decreasing conﬂuent hypergeometric function

using [[28], Eq. (1.6)]. Consequently, these inﬁnite series terms

converge rapidly as we discussed earlier in Table I.

C. SC with Triple Correlated Branches

In this section, we consider triple correlated diversity

branches. We start from PDF of the fading envelope for

trivariate Nakagami-

m

channels given in [[26], Eq. (8)]. Then,

by changing variable and by assuming identical branches

(

¯γ= ¯γ1= ¯γ2= ¯γ3

, and the same fading parameter

m

), the

PDF of the output SNR for triple SC exponentially correlated

Nakagami-mbranches can be derived. This is shown below

pγ SC,3(γ) = Σ−1m

Γ(m)

∞

X

i=0

∞

X

j=0

|p1,2|2i|p2,3|2j

pi+m

1,1pi+j+m

2,2pj+m

3,3

×[Θ1+ Θ2+ Θ3]

Γ(m+i) Γ(m+j)i!j!,

(20)

where

Σ−1

is the inverse of the correlation matrix,

pi1,j1(i1, j1= 1,2,3)

being its entries and

Θ1,Θ2

and

Θ3

are

Θ1=p1,1m

¯γi+m

γi+m−1e−p1,1m

¯γγ

×γi+j+m, p2,2m

¯γγγj+m, p3,3m

¯γγ,

(21)

Θ2=p2,2m

¯γi+j+m

γi+j+m−1e−p2,2m

¯γγ

×γi+m, p1,1m

¯γγγj+m, p3,3m

¯γγ,

(22)

Θ3=p3,3m

¯γj+m

γj+m−1e−p3,3m

¯γγ

×γi+m, p1,1m

¯γγγi+j+m, p2,2m

¯γγ,

(23)

respectively. Here

γ(a, x)

denotes the lower incomplete

gamma function with

γ(a, x) = Rx

0e−tta−1dt

([18],

Eq.(8.350/1)).

In exponentially correlated model, the diversity antennas are

equispaced. Therefore, the correlation matrix can be written as

Σi1,j1≡ρ|i1−j1|

[27]. Hence, the inverse correlation matrix

Σ−1is tridiagonal and can be written as

Σ−1=1

ρ2−1

−1ρ0

ρ−(ρ2+ 1) ρ

0ρ−1

,(24)

where ρdenotes the correlation coefﬁcient.

We have made an assumption of identical average SNRs in

all three branches above. This assumption is reasonable if the

diversity channels are closely spaced and, their gains as well

as noise powers are equal [3].

The average probability of detection for triple SC diversity

Nakagami-

m

correlated branches with integer

u

is derived as

below. See Appendix D for details.

PD,S C,3=Σ−1m

Γ(m)e−λ

2

∞

X

n=0

n+u−1

X

k=0

∞

X

i=0

∞

X

j=0

"λ

2k¯γ

mn|p1,2|2i|p2,3|2j

pi+m

1,1pi+j+m

2,2pj+m

3,3Γ(m+i)Γ(m+j)

×pi+m

1,1pi+j+m

2,2pj+m

3,3Γ(2i+ 2j+ 3m+n)

p11 +p2,2+p3,3+¯γ

m(2i+2j+3m+n)i!j!k!n!

×(Ξ1+ Ξ2+ Ξ3)#.(28)

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 5

Ξ1=

F22i+ 2j+ 3m+n; 1,1; i+j+m+ 1, j +m+ 1; p2,2

p11 +p2,2+p3,3+¯γ

m

,p3,3

p11 +p2,2+p3,3+¯γ

m

(i+j+m) (j+m),(25)

Ξ2=

F22i+ 2j+ 3m+n; 1,1; i+m+ 1, j +m+ 1; p1,1

p11 +p2,2+p3,3+¯γ

m

,p3,3

p11 +p2,2+p3,3+¯γ

m

(i+m) (j+m),(26)

Ξ3=

F22i+ 2j+ 3m+n; 1,1; i+m+ 1, i +j+m+ 1; p1,1

p11 +p2,2+p3,3+¯γ

m

,p2,2

p11 +p2,2+p3,3+¯γ

m

(i+m) (i+j+m).(27)

Here,

Ξ1,Ξ2

and

Ξ3

are as given in (25), (26) and (27) at the

top of next page, respectively, and

F2α3;β3, β0

3;γ3, γ0

3;x, y

denotes the Hypergeometric function of two variables deﬁned

in [[18], Eq. (9.180.2)]. Note, for

m= 1

, (28) reduces to triple

correlated Rayleigh fading branches.

D. General Expression for Triple Branches

In this section, we will derive a general and simpler

alternative expression to (28), where both

u

and

m

are not

restricted to integer values. See Appendix E for details.

The average probability of detection for triple SC

Nakagami-

m

correlated branches for not restricted

u

or

m

integer values is:

PD,S C,3=Σ−1m

Γ(m)

∞

X

n=0

∞

X

i=0

∞

X

j=0 ¯γ

mnΓu+n, λ

2

Γ(u+n)

×|p1,2|2i|p2,3|2jΓ(2i+ 2j+ 3m+n)

Γ(m+i) Γ(m+j)i!j!n!

×(Ξ1+ Ξ2+ Ξ3)

p11 +p2,2+p3,3+¯γ

m2i+2j+3m+n.(29)

where

Ξ1,Ξ2

and

Ξ3

are given in (25), (26) and (27),

respectively. As before, for

m= 1

, (29) reduces to triple

correlated Rayleigh fading branches. It’s worthwhile to

mention that the Hypergeometric function of two variables

F2α3;β3, β0

3;γ3, γ0

3;x, y

appears in (28) and (29) converges

only for

|x|+|y|<1

[18], where

|.|

denotes absolute.

Fortunately, this is the case in our above derived equations.

IV. DUAL CORRELATE D NAKAGAMI-mCHANNELS WITH

SSC DIVERSITY

The SSC receiver selects a particular diversity branch

until its SNR drops below a predetermined threshold value.

Hence SSC’s technique is similar to its counterpart SC but.

Nevertheless, the SSC receive does not need to continuously

monitor the SNR of each branch. Therefore, the SSC

is considered as the least complex

2

diversity combining

technique [3].

2

Other diversity combining techniques such EGC and MRC process more

than one branch and require the channel state knowledge of some or all the

branches [3].

Starting from [[3], p.437, Eq. (9.334)], the SNR’s PDF for

a dual and identical correlated Nakagami-

m

fading channels

with SSC combiner is

pγ SS C (γ) = (A(γ)γ≤γT

A(γ) + m

¯γmγm−1

Γ(m)exp −m γ

¯γγ > γT,

(30)

where

γT

denotes a predetermined switching threshold and

A(γ)is given in [[3], p.437, Eq. (9.335)] as

A(γ) = m

¯γmγm−1

Γ(m)exp −m γ

¯γ

×h1−Qmp2a ρ γ, p2a γTi,

(31)

where

a=m

¯γ(1−ρ)

and

Qm(., .)

denotes generalized Marcum

Q-function.

The average probability of detection for dual correlated

Nakagami-mfading branches with SSC diversity (PD,SS C,2)

is obtained by substituting (30) into (9) and then using the

deﬁnition R∞

afdx=R∞

0fdx−Ra

0fdx, which yields

PD,S SC,2=1

Γ(m)m

¯γm

[IA−IB−IC](32)

with

IA= 2 Z∞

0

Qup2γ, √λγm−1exp −m γ

¯γdγ, (33)

IB=Z∞

0

Qup2γ, √λQmp2a ρ γ , p2a γT

×γm−1exp −m γ

¯γdγ, (34)

and

IC=ZγT

0

Qup2γ, √λγm−1exp −m γ

¯γdγ. (35)

Before deriving an expression for the probability of detection

PD,S SC,2

, it is worthy to investigate (32) for the following

two special cases of threshold values.

Case I: γT= 0

If

γT= 0

, we have

Qm√2a ρ γ, √2a γT= 1

and the third

term

IC

vanishes, consequently (32) reduces to single branch

detection as

PD,S SC,2=1

Γ(m)m

¯γmZ∞

0

Qup2γ, √λ

×γm−1exp −m γ

¯γdγ.

(36)

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 6

Case II: γT→ ∞

If

γT→ ∞

, we have

Qm√2a ρ γ, √2a γT= 0

,

consequently

IB

vanishes and only

IC

is subtracted from

IA

.

This results in single branch detection as in (36). Therefore, care

must be taken to choose a sensible threshold value. Otherwise,

the diversity technique might become useless.

The average probability of detection for dual correlated SSC

receiver with Nakagami-

m

fading branches where

u

and

m

are restricted to integer values is given in (37) at the top of

next page. Please see Appendix F for detailed derivation.

Note that for

m= 1

, (37) reduces to dual Rayleigh correlated

fading branches, and for

ρ= 0

it reduces to dual

i.i.d.

Nakagami-mfading branches detection.

A. Alternative Solution

The expression

PD,S SC,2

in (37) involves many inﬁnite

series representations. Some of their upper bounds (number of

terms) are dependent on the preceded one. As an example the

upper bound of the second sum (

Pj+u−1

k=0 (.)

) depends on the

number of terms

(N)

needed for convergence of the previous

series. Fortunately, it will not be very difﬁcult to ﬁnd the

number of terms for convergence (with ﬁve digit accuracy).

However, time for numerical implementation will be rather

long. Therefore, we will derive an alternative more general

and simpler expression

PD,S SC,2

with less number of inﬁnite

series representations.

The average probability of detection where

u

is not restricted

while (

m≥1

) is restricted to integer values is given in (38)

at the top of next page. See Appendix G for the derivation.

Note, for

m= 1

, (38) reduces to that of a dual SSC receiver

with Rayleigh correlated fading branches. For,

ρ= 0

it reduces

to the PDF of the dual

i.i.d.

Nakagami-

m

fading branches

detection.

Interestingly, the three terms in (38) contain the upper

incomplete gamma function in addition to the lower incomplete

gamma function in the last term. In fact, we can represent

both these functions by the monotonically decreasing conﬂuent

hypergeometric function using [[19], Eq. (6.5.12)] and [[28],

Eq. (1.6)] for lower and upper incomplete gamma functions,

respectively. Consequently the inﬁnite series terms in (38)

converges rapidly.

B. Optimal Threshold (γ∗

T)

Optimal threshold

γ∗

T

is deﬁned as the value of the SNR

that maximizes the probability of detection. We maximize the

probability of detection by selecting an appropriate SNR for

SSC switching. Probability of false alarm is ﬁxed since it’s

a function of the decision threshold

λ

and not a function of

SNR, as shown in (7). Constant False Alarm Rate (CFAR) is

a well-known technique that is often employed in cognitive

spectrum sensing. In this technique and using (7), a decision

threshold is calculated for ﬁxed probability of false alarm. Then

the corresponding probability of detection is calculated using

(8) for optimal SNR. We have derived an expression for this

optimal threshold given in (39) at the top of this page. This is

done by differentiating

PD,S SC,2

in (32) with respect to

γT

and solving

∂

∂γ∗

TPD,S SC,2= 0

for

γ∗

T

. See Appendix H for

Table I: Terms required for ﬁve digits accuracy

PD,S C,2:

˜

EN

, u = 2, P F= 0.01,¯γ= 20 dB

ρm= 1

Nn,Ni

m= 2

Nn,Ni

m= 3

Nn,Ni

m= 4

Nn,Ni

0 15,1 15,1 15,1 15,1

0.2 15,3 15,3 15,2 15,1

0.4 15,3 15,2 15,1 15,1

0.6 15,3 15,5 15,4 15,4

0.8 15,4 15,5 15,6 15,7

details. Using Matlab, we can obtain the optimal threshold by

evaluating (39) numerically for ∂

∂γ∗

TPD,S SC,2= 0.

V. SIMULATION AND ANALYS IS RE SU LTS

The energy detector employed in spectrum sensing is

mainly characterized by the probability of false alarm

PF

and

probability of detection

PD

. In this section we study the impact

of the correlation among antenna diversity branches on

PD

(equivalently probability of miss detection

PDm = 1 −PD

) as

a performance metric using the derived expressions in previous

sections. To this end, we produce Complementary Receiver

Operating Characteristic (CROC) graphs (

PDm

versus

PF

)

for SC and SSC diversity techniques in Nakagami-

m

fading

channel.

First, we plot the probability of miss detection with the

corresponding threshold

λ

for

u= 2

,

¯γ= 20

dB,

m∈(1,4)

and,

ρ∈(0−0.8)

for different values of

PF

using (7). Through

Monte Carlo simulation, we obtain the CROC curves for SC

and SSC. We then compare the simulation results with the

analytical curves obtained from derived expressions.

In Figure. 1, we plot the CROC graphs for

L= 2, m =

1,¯γ= 20

dB and

ρ= 0.8

. Results are obtained for SC and

SSC using both the derived expressions (analytical) and by

Monte Carlo simulation. For SC diversity, both these curves

are almost in a perfect match. However, reader may observe a

very small difference between analytical and simulation curves

for SSC diversity. This is due to the inaccuracy arising from

rounding off the inﬁnite series and calculating the optimal

threshold .

In Figure 2, we plot the CROC graphs for SC with

¯γ=

20

dB,

m∈(1,4)

and,

ρ∈(0 −0.8)

. For each value of

fading severity

m

, one can clearly notice the degradation in the

probability of detection due to the correlation among diversity

branches. For instance, let us consider the case

m= 1

and

constant

PF= 0.01

as in Figure 2a. The corresponding

PDm

for

ρ= 0.8

is almost four times its value for

ρ= 0

(no

correlation). Similar result could be observed in Figure 2b,

however, the increment ratio is now much more larger. However,

as

m

increases (low fading environment), correlation effect is

compensated for, resulting in higher probability of detection

(equivalently, low probability of miss detection). Thus, the rate

of correlation compensation due to good channel is higher than

the correlation impact on probability of detection.

For easy and better comparison between SC and SSC and

their performance in combating the correlation, we plot CROC

graphs in Figure 3 for

¯γ= 20

dB,

m∈(1,4)

and,

ρ∈(0,0.8)

.

As before, one can notice the impact of the correlation between

fading branches on the probability of detection. This impact is

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 7

PD,S SC,2=1

Γ(m)m

¯γm

e−λ

2"2

∞

X

j=0

j+u−1

X

k=0 λ

2k(j+m−1)!

j!k!¯γ

¯γ+mj+m

−

∞

X

n=0

n+u−1

X

q=0

∞

X

i=0

i+m−1

X

k=0 λ

2q

×ai+kρi(m+n+i−1)!

a ρ +m

¯γ+ 1m+n+ii!k!n!q!

e−a γTγk

T−

∞

X

n=0

n+u−1

X

q=0 λ

2q1

n!q!¯γ

¯γ+mm+n

γm+n, γT¯γ+m

¯γ#(37)

PD,S SC,2=1

Γ(m)m

¯γm"4

∞

X

j=0

Γu+j, λ

2Γ(m+j)

Γ(u+j)1 + m

¯γm+jj!−

∞

X

n=0

∞

X

i=0

i+m−1

X

k=0

Γu+n, λ

2Γ(m+n+i)γk

Te−aγTai+kρi

Γ(u+n)m

γ+aρ + 1m+n+in!i!k!

−

∞

X

p=0

Γu+p, λ

2

Γ(u+p)p!¯γ+m

¯γ−(m+p)

γm+p, γT¯γ+m

¯γ#.

(38)

∂

∂γ∗

T

PD,S SC,2=1

Γ(m)m

¯γm"p2a γ∗

Te−aγ∗

T

∞

X

k=0

am+2k−1ρk

Γ(m+k)2m+kk!γ∗

T

k(G0

1+1

2

u−1

X

n=1 λ

2nΓ (m+k)

a+1

2m+kn!

×1F1m+k;n+ 1; λ

2 (a+ 1))−Qup2γ∗

T,√λγ∗

T

m−1exp −m γ∗

T

¯γ#.

(39)

10-4 10-3 10-2 10-1 100

Probabilty of False Alarm PF

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

Probability of Miss Detection PDm

SC Analytic

SC Simulation

SSC Analytic

SSC Simulation

Figure 1: Analytic (solid) versus simulation (dashed) results for

SC and SSC derived expressions with

L= 2, m = 1

,

¯γ= 20

dB and ρ= 0.8.

compensated by good channel. Furthermore, results in Figure 3

show that SC outperforms SSC. This is a well proven fact in the

literature. In fact, performance difference is more pronounced

for uncorrelated (

ρ= 0

) and high

m

values. However, we may

notice that as the correlation increases between the branches,

the performance of both SC and SSC schemes becomes more

comparable. This is especially true for high mvalues.

Figure 4 shows probability of miss detection versus

correlation for

¯γ= 20

dB,

m∈(1,4)

and,

PF= 0.01

for both

SC and SSC diversity techniques. Another interesting behaviour

that could be observed from this ﬁgure. As

m

increases

(equivalently, fading decreases), less signiﬁcant deterioration

in probability of detection is observed due to correlation. In

other words, the loss in diversity gain due to correlation gets

lower as mincreases.

To gain better insight about this behaviour, let us discuss it

with more details. Figure 4a shows clearly this interesting

behaviour. The curve for

m= 1

in Figure 4a has an

average high positive slope. Consequently, the probability

of detection degrades rapidly as correlation increases. As

m

increases, corresponding curves get ﬂattened (slope decreases).

Consequently, probability of detection degrades slowly as

correlation increases. This can be attributed to the fact that

already the

PD

values are high due to low fading. On the other

hand, for small

m

-values (deep fading), correlation signiﬁcantly

deteriorates the probability of detection which is already poor.

A similar behaviour could be observed in the SSC shown in

Figure 4b. Therefore, we conclude the following. In a deep

fading scenario, the inter-branch correlation is a crucial factor

and its effects must be incorporated in any spectrum sensing

model. By contrast, in a low fading environment (those having

large values of

m

), the effect of such correlation may be ignored

without much impact.

VI. CONCLUSION

In this work, we have investigated the impact of correlation

among diversity fading branches in multi-antenna cognitive

radio spectrum sensing networks. A uniﬁed performance

analysis was presented for the probability of detection of

SC and SSC diversity combining receivers with arbitrary

and exponential correlation among fading branches. Exact

expressions were derived for the probability of detection for

each case. Our result show that the correlation among diversity

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 8

(a) m= 1 (Rayleigh)

10-4 10-3 10-2 10-1 100

Probability of False Alarm "PF"

10-14

10-12

10-10

10-8

10-6

10-4

10-2

Probability of Miss Detection " PDm"

= 0

= 0.2

= 0.4

= 0.6

= 0.8

(b) m= 4

Figure 2: SC dual correlated Nakagami-mbranches with ¯γ= 20 dB for different ρvalues.

10-4 10-3 10-2 10-1 100

Probability of False Alarm "PF"

10-14

10-12

10-10

10-8

10-6

10-4

10-2

Probability of Miss Detection " PDm"

SC: = 0

SC: = 0.8

SSC: = 0

SSC: = 0.8

(a) m= 1 (Rayleigh)

10-4 10-3 10-2 10-1 100

Probability of False Alarm "PF"

10-14

10-12

10-10

10-8

10-6

10-4

10-2

Probability of Miss Detection " PDm"

SC: = 0

SC: = 0.8

SSC: = 0

SSC: = 0.8

(b) m= 4

Figure 3: SC/SSC dual correlated Nakagami-mbranches comparison with ¯γ= 20 dB for ρ= 0 (solid) and 0.8 (dashed).

fading branches causes an adverse impact on the probability

of detection, which cannot be ignored especially under severe

fading conditions. Consequently, an increase in the interference

rate between the primary user and secondary user is observed

by three times its rate when independent fading branches

is assumed. Our investigations reveal that for low fading

environment (large

m

-values), correlation effect may be ignored.

Furthermore, at low fading and highly correlated environments,

SSC which is simpler scheme performs as good as SC which

is a more complex scheme.

APPENDIX A

DERIVATION OF (12)

Using ([17], (20)), the PDF of SC’s output of dual identical

correlated Nakagami fading branches is

pγ SC (r) = 4mmr2m−1

Γ(m)Ωmexp −m r2

Ω

×h1−Qmp2ρA r, √2A ri,

(40)

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Correlation " "

0

2x10-3

4x10-3

6x10-3

8x10-3

10x10-3

12x10-3

14x10-3

16x10-3

18x10-3

Probability of Miss Detection " PDm"

m = 1

m = 2

m = 3

m = 4

(a) SC

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Correlation " "

0

5x10-3

10x10-3

15x10-3

20x10-3

25x10-3

Probability of Miss Detection " PDm"

m = 1

m = 2

m = 3

m = 4

(b) SSC

Figure 4: Probability of miss detection versus correlation with

¯γ= 20

dB,

PF= 0.01

and different fading severity for SC and

SSC.

where A=qm

Ω(1−ρ).

Changing variables using pγ(γ) = prqΩγ

¯γ

2√¯γ γ

Ω[3] yields

pγ SC (γ) =

4mmqΩγ

¯γ2m−1

2q¯γ γ

ΩΓ(m)Ωm

exp

−

mqΩγ

¯γ2

Ω

×"1−Qm p2ρAsΩγ

¯γ,√2AsΩγ

¯γ!#,

(41)

Simplifying, (41) becomes

pγ SC (γ) = sΩ

¯γ γ !2mmqΩγ

¯γ2m−1

Γ(m)Ωmexp −m γ

¯γ

×"1−Qm p2ρAsΩγ

¯γ,√2AsΩγ

¯γ!#,

(42)

Substituting A=qm

Ω(1−ρ)and simplifying, yields

pγ SC (γ) = Ω1/2+m−1/2

¯γ1/2γ1/2

2mmγm−1/2

Γ(m)¯γm−1/2Ωmexp −m γ

¯γ

"1−Qm p2ρrm

Ω(1 −ρ)sΩγ

¯γ,√2rm

Ω(1 −ρ)sΩγ

¯γ!#,

(43)

Simplifying

pγ SC (γ) = 2mm

Γ(m)¯γmγ1−mexp −m γ

¯γ

×"1−Qm s2mρ

¯γ(1 −ρ)γ, s2m

¯γ(1 −ρ)γ!# (44)

Simplifying and rearranging, this concludes the derivation.

APPENDIX B

EXPRESSION FOR DUAL S C

In this appendix, we derive the expression in (16).

1) Evaluating

IA

in (14): Introducing changing variable

x=√2γ, we can derive

IA=1

2m−1Z∞

0

Qux, √λx2m−1exp −m x2

2 ¯γdx

| {z }

I

.

(45)

Using [[29], Eq. (29)], we write

Z∞

0

Qu(α x, β )xqe−p2x2

2dx≡Gu

=Gu−1+

Γq+1

2β2

2u−1e−β2

2

2 (u−1)! p2+α2

2q+1

2

×1F1q+ 1

2;u;β2

2

α2

p2+α2, q > −1,(46)

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 10

we can solve

I

by evaluating

Gu

recursively for

q > −1

and

restricted uinteger values as

Gu=Gu−1+Au−1Fu

=Gu−2+Au−2Fu−2+Au−1Fu−1

.

.

.

=G1+

u−1

X

n=1

AnFn+1.

(47)

where Anand Fnare given as

An=1

2 (n!) p2+α2

2q+1

2

Γq+ 1

2β2

2n

e−β2

2,(48)

Fn=1F1q+ 1

2;n;β2

2

α2

p2+α2,(49)

and

1F1(., .;.)

as deﬁned previously in (18). Hence, we solve

(45) to obtain

IA=1

2m−1G1+η

2

u−1

X

n=1

1

n!λ

2n

×1F1m;n+ 1; λ¯γ

2 (m+ ¯γ),

(50)

where

η= Γ(m)2 ¯γ

m+¯γme−λ

2

and

G1

can be obtained by

evaluating the following integral containing the ﬁrst order of

Marcum Q-function Q(., .)for integer mvalues as

G1=Z∞

0

Qx, √λx2m−1e−m x2

2 ¯γdx. (51)

Using [[29], Eq. (25)], we evaluate

G1

for integer

m

values

as in (17).

2) Evaluating

IB

in (15): Using the alternative canonical

Marcum Q-function representations for

Qu√2γ, √λ

given

in [15] for not restricted to integer values of uas

Qup2γ, √λ=

∞

X

n=0

γne−γΓu+n, λ

2

Γ(u+n)n!,(52)

and the alternative representation given in [[3], Eq. (4.74)] for

restricted minteger values as

Qm(α1, β1) =

∞

X

i=0

exp −α2

1

2α2

1

2i

i!

×

i+m−1

X

k=0

exp −β2

1

2β2

1

2k

k!,

(53)

therefore, Qm√2a ρ γ, √2a γcould be written as

Qmp2aργ, p2aγ=

∞

X

i=0

i+m−1

X

k=0

ai+kρi

i!k!e−aγ(ρ+1) γi+k.

(54)

Then by substituting (52) and (54) into (15) with some

simpliﬁcation we derive

IB=

∞

X

n=0

∞

X

i=0

i+m−1

X

k=0

Γu+n, λ

2

Γ(u+n)n!

ai+kρi

i!k!

×Z∞

0

γi+k+m+n−1e−γc dγ

| {z }

I

,

(55)

where

c= 1 + a(ρ+ 1) + m

¯γ

. Now the next task is solving

the integral

I

in (55). For this we use [[18], Eq. (3.351/3)] and

satisfying the condition therein,

Z∞

0

xp1e−µ1xdx=p1!µ1−p1−1[Re µ1>0] .(56)

Hence (55) becomes

IB=

∞

X

n=0

∞

X

i=0

i+m−1

X

k=0

Γu+n, λ

2

Γ(u+n)

(i+k+m+n−1)!ai+kρi

ci+k+m+nn!i!k!

(57)

Substituting (50) and (57) into (13), this concludes the

derivation.

APPENDIX C

ALTER NATI VE EXPRESSION FOR DUA L SC

In this Appendix, we derive (19). Using the alternative

expression for Marcum Q-function given in [[3], Eq.(4.63)],

where

u

is not restricted to integer values, we can write

Qu√2γ, √λas

Qup2γ , √λ= 1 −e−2γ+λ

2

∞

X

n=u √λ

√2γ!n

Inp2λγ.

(58)

Then substituting (12) in (9) and using the deﬁnition of the

PDF as

Z∞

0

pγ(γ) dγ= 1,(59)

with simpliﬁcation, we can derive

PD,S C,2= 1 −[IA−IB],(60)

where

IA=2

Γ(m)m

¯γmZ∞

0

γm−1e−2γ+λ

2

∞

X

n=u √λ

√2γ!n

×Inp2λ γ exp −m γ

¯γdγ,

(61)

and

IB=2

Γ(m)m

¯γmZ∞

0

γm−1e−2γ+λ

2

∞

X

n=u √λ

√2γ!n

×Inp2λγexp −mγ

¯γQmp2aργ, p2aγdγ, γ ≥0.

(62)

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 11

1) Evaluating

IA

in (61): Simplifying and rearranging (61),

we derive

IA=2

Γ(m)m

¯γm

e−λ

2

∞

X

n=uλ

2n

2

×Z∞

0

γm−n

2−1e−γ(1+ m

¯γ)Inp2λ γ dγ.

(63)

Using [[18], Eq. (6.643/2)] given as

Z∞

0

xµ−1

2e−α x I2ν2β√xdx=

Γµ+ν+1

2

Γ(2 ν+ 1) β−1eβ2

2αα−µM−µ,ν β2

α,

Re µ+ν+1

2>0,(64)

where Mµ,ν (.)denotes Whittaker function given by [18]

Mµ,ν (z) = zν+1

2e−z

21F1ν−µ+1

2; 1 + 2ν;z,(65)

with some simpliﬁcation and rearranging, the solution of (63)

can be derived as

IA=2

dmm

¯γm

e−λ

2

∞

X

n=uλ

2n1

Γ(n+ 1)

×1F1m; 1 + n;λ

2d,

(66)

where d=¯γ+m

¯γ.

2) Evaluating

IB

in (62): Simplifying and rearranging (62),

we derive

IB=2

Γ(m)m

¯γm

e−λ

2

∞

X

n=uλ

2n

2Z∞

0

γm−n

2−1e−d γ

×Inp2λ γ Qmp2a ρ γ, p2a γ dγ.

(67)

Using (54) with simpliﬁcation and rearranging, we write (67)

as

IB=2

Γ(m)m

¯γm

e−λ

2

∞

X

n=u

∞

X

i=0

i+m−1

X

k=0 λ

2n

2ai+kρi

i!k!

×Z∞

0

γm−n

2+i+k−1e−γc dγ,

(68)

where

c= 1 + a(ρ+ 1) + m

¯γ

. Similarly, implementing same

procedures as (66), the solution of (68) can be given as

IB=2

Γ(m)m

¯γm

e−λ

2

∞

X

n=u

∞

X

i=0

i+m−1

X

k=0 λ

2nai+kρi

i!k!

×Γ(m+i+k)

Γ(n+ 1) cm+i+k1F1m+i+k; 1 + n;λ

2c.

(69)

Substituting (66) and (69) into (60), this concludes the

derivation.

APPENDIX D

EXPRESSION FOR TRIPLE SC

In this section, we drive (28). Using (53) and substituting

(20) into (9), we drive the average probability of detection as

PD,S C,3=Σ−1m

Γ(m)e−λ

2

∞

X

n=0

n+u−1

X

k=0

∞

X

i=0

∞

X

j=0

×|p1,2|2i|p2,3|2jλ

2k

pi+m

1,1pi+j+m

2,2pj+m

3,3Γ(m+i) Γ(m+j)i!j!k!n!

×Z∞

0

γne−γ[Θ1+ Θ2+ Θ3] dγ

| {z }

IA

.

(70)

Substituting (21), (22) and (23) into (70), the integral part

IA

in (70) becomes

IA=p1,1m

¯γi+m

Ia1+p2,2m

¯γi+j+m

Ia2

+p3,3m

¯γj+m

Ia3,

(71)

where

Ia1=Z∞

0

γi+m+n−1e−γ(p11 m

¯γ+1)(72)

×γi+j+m, p2,2m

¯γγγj+m, p3,3m

¯γγdγ.

Ia2=Z∞

0

γi+j+m+n−1e−γ(p2,2m

¯γ+1)(73)

×γi+m, p1,1m

¯γγγj+m, p3,3m

¯γγdγ.

Ia3=Z∞

0

γj+m+n−1e−γ(p3,3m

¯γ+1)(74)

×γi+m, p1,1m

¯γγγi+j+m, p2,2m

¯γγdγ.

Each integral in (71) could be written as

I=Z∞

0

xae−bxγ(d1, c1x)γ(d2, c2x) dx. (75)

Using [[30], Eq. (10)], we write (71) as

IA=¯γ

mnpi+m

1,1pi+j+m

2,2pj+m

3,3Γ(2i+ 2j+ 3m+n)

p11 +p2,2+p3,3+¯γ

m(2i+2j+3m+n)

×(Ξ1+ Ξ2+ Ξ3),

(76)

where

Ξ1

,

Ξ2

and

Ξ3

are in (25), (26) and (27), respectively.

Substituting (76) into (70), this concludes the derivations.

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 12

APPENDIX E

GEN ER AL EXPRESSION FOR TRIPLE SC

In this section, we derive the expression in (29). Using (52)

and substituting (20) into (9) we derive

PD,S C,3=Σ−1m

Γ(m)

∞

X

n=0

∞

X

i=0

∞

X

j=0

Γu+n, λ

2

Γ(u+n)

×|p1,2|2i|p2,3|2j

pi+m

1,1pi+j+m

2,2pj+m

3,3Γ(m+i) Γ(m+j)i!j!n!

×Z∞

0

γne−γ[Θ1+ Θ2+ Θ3] dγ

| {z }

IA

.(77)

Following same procedures in (71)-(76), then substituting (76)

into (77), this concludes the derivation.

APPENDIX F

EXPRESSION FOR DUAL SSC

In this section, we will derive the expression in (37) by

evaluating PD,SS C,2in (32) as follows.

1) Integral

IA

in (33): Using Marcum Q-function alternative

representation (53), we rewrite (33) as

IA= 2 e−λ

2

∞

X

j=0

j+u−1

X

k=0 λ

2k1

j!k!

×Z∞

0

γj+m−1exp −γ(1 + m

¯γ)dγ.

(78)

Using (56) and satisfying the condition therein, we solve (78)

as

IA= 2 e−λ

2

∞

X

j=0

j+u−1

X

k=0 λ

2k(j+m−1)!

j!k!¯γ

¯γ+mj+m

.

(79)

2) Integral

IB

in (34): Following the same procedures as

in (78), we rewrite (34) as

IB=e−λ

2

∞

X

n=0

j+u−1

X

q=0

∞

X

i=0

i+m−1

X

k=0 λ

2qai+kρi

i!k!n!q!e−a γTγk

T

×Z∞

0

γm+n+i−1exp −γa ρ +m

¯γ+ 1dγ.

(80)

Similarly as we did in (79), we solve (80) as

IB=e−λ

2

∞

X

n=0

j+u−1

X

q=0

∞

X

i=0

i+m−1

X

k=0 λ

2qai+kρi

i!k!n!q!

×(m+n+i−1)!

a ρ +m

¯γ+ 1m+n+ie−a γTγk

T.

(81)

3) Integral ICin (35): Using (53), we rewrite (35) as

IC=e−λ

2

∞

X

n=0

n+u−1

X

q=0

1

n!q!λ

2k

×ZγT

0

γm+n−1exp −γm

¯γ+ 1dγ.

(82)

Using [[18], Eq. (3.351/1)], where

Zz

0

xne−µ xdx=n!

µn+1 −e−µz

n

X

k=0

n!

k!

zk

µn−k+1

=µ−n−1γ(n+ 1, µz),

[z > 0,Re µ > 0, n = 0,1,2,···],

(83)

we derive (82) as

IC=e−λ

2

∞

X

n=0

n+u−1

X

q=0

1

n!q!λ

2q¯γ

¯γ+mm+n

×γm+n, γT¯γ+m

¯γ.

(84)

Substituting (79), (80) and (84) into (32), this concludes the

derivation.

APPENDIX G

ALTER NATI VE EXPRESSION FOR DUA L SSC

In this section, we will derive the expression in (38) by

evaluating

PD,S SC,2

in (32) for alternative expression as

follows.

1) Integral

IA

in (33): Let

x=√2γ

, we rewrite (33) as

IA=4

2−mZ∞

0

Qux, √λxm−1exp −m x2

2¯γdx. (85)

Using [[31], Eq. (8)], we solve (85) as

IA= 4

∞

X

j=0

Γ(m+j)Γu+j, λ

2

Γ(u+j)1 + m

¯γm+jj!

.(86)

2) Integral

IB

in (34): Using (52) and (53) for

Qu√2γ, √λ

and

Qm√2a ρ γ, √2a γT

, we rewrite (34)

as

IB=

∞

X

n=0

∞

X

i=0

i+m−1

X

k=0

Γu+n, λ

2

Γ(u+n)n!

ai+kρi

i!k!γk

Te−a γT

×Z∞

0

γm+n+i−1e−γ(m

¯γ+a ρ+1)dγ.

(87)

Using (56) , we solve (87) as

IB=

∞

X

n=0

∞

X

i=0

i+m−1

X

k=0

Γu+n, λ

2Γ(m+n+i)

Γ (u+n)m

γ+aρ + 1m+n+in!i!k!

×ai+kρiγk

Te−aγT.

(88)

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 13

3) Integral ICin (35): Using (52), we rewrite (35) as

IC=

∞

X

p=0

Γu+p, λ

2

Γ(u+p)p!ZγT

0

γm+p−1e−γ(¯γ+m

¯γ)dγ. (89)

Using (83), we solve (89) for minteger values as

IC=

∞

X

p=0

Γu+p, λ

2

Γ(u+p)p!¯γ+m

¯γ−(m+p)

×γm+p, γT¯γ+m

¯γ.

(90)

Substituting (86), (88) and (90) into (32), this concludes the

derivation.

APPENDIX H

EXPRESSION FOR OPTIMAL THR ES HO LD

In this section, we will derive the expression in (39).

Employing Leibniz’s rule [[19], Eq. (3.3.7)] with the aid of

following identity given in [[29], Eq. (9)] as

∂

∂β Qu(α, β ) = −ββ

αu−1

exp −α2+β2

2Iu−1(α β),

(91)

we rewrite (32) as

∂

∂γ∗

T

PD,S SC,2=1

Γ(m)m

¯γm"ρ1−m

2√2a γ∗

T

1−m

2e−aγ∗

T

×Z∞

0

Qup2γ, √λγm−1

2e−aγ Im−12apργ∗

Tγdγ

| {z }

I

−Qup2γ∗

T,√λγ∗

T

m−1exp −m γ∗

T

¯γ#.

(92)

To solve the integral

I

in (92), we perform changing variable

along with the aid of the series expansion of the modiﬁed

Bessel function given in [[18], Eq. (8.445)] as

Iν(z) =

∞

X

k=0

1

Γ(ν+k+ 1) k!z

2ν+2 k.(93)

Then, we drive (92) as

∂

∂γ∗

T

PD,S SC,2=1

Γ(m)m

¯γm"ρ1−m

2p2a γ∗

Te−aγ∗

T

×

∞

X

k=0

1

Γ(m+k)2m+kk!(a√ρ)m+2k−1γ∗

T

k

×Z∞

0

Qux, √λx2(m+k)−1e−a

2x2dx

| {z }

I

−Qup2γ∗

T,√λγ∗

T

m−1exp −m γ∗

T

¯γ#.

(94)

Using [[29], Eq. (29)] by following same procedures as in (50),

we can solve the integral Iin (94) as

IA=G0

1+1

2

u−1

X

n=1 λ

2nΓ (m+k)

a+1

2m+kn!

×1F1m+k;n+ 1; λ

2 (a+ 1),

(95)

where

G0

1

can be obtained by evaluating the following integral

containing the ﬁrst order of Marcum

Q

-function

Q(., .)

for

integer mvalues as

G0

1=Z∞

0

Qx, √λx2(m+k)−1e−a

2x2dx. (96)

Using [[29], Eq. (25)], we evaluate

G0

1

for integer

m

values as

G0

1=2m+k−1(m+k−1)!

a2(m+k)1

a+ 1e−λ

2

a

a+1

×"(1 + a)a

1 + am+k−1

Lm+k−1−λ

2 (1 + a)

+

m+k−2

X

n=0 a

a+ 1n

Ln−λ

2 (a+ 1)#,

(97)

where Ln(.)denotes Laguerre polynomial of n-degree [18].

Substituting (95) into (94), this concludes the derivation.

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